Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $q \neq 0$. $p = \dfrac{-5}{3(q + 9)} \div \dfrac{5q}{9q + 81} $
Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{-5}{3(q + 9)} \times \dfrac{9q + 81}{5q} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ -5 \times (9q + 81) } { 3(q + 9) \times 5q } $ $ p = \dfrac {-5 \times 9(q + 9)} {5q \times 3(q + 9)} $ $ p = \dfrac{-45(q + 9)}{15q(q + 9)} $ We can cancel the $q + 9$ so long as $q + 9 \neq 0$ Therefore $q \neq -9$ $p = \dfrac{-45 \cancel{(q + 9})}{15q \cancel{(q + 9)}} = -\dfrac{45}{15q} = -\dfrac{3}{q} $